CONTROLLABILITY OF THE ONE-DIMENSIONAL FRACTIONAL HEAT

EQUATION UNDER POSITIVITY CONSTRAINTS

UMBERTO BICCARI1,2, MAHAMADI WARMA3, AND ENRIQUE ZUAZUA1,2,4,5

Abstract. In this paper, we analyze the controllability properties under positivity constraints on the controlor the state of a one-dimensional heat equation involving the fractional Laplacian (−d 2

x )s (0 < s < 1) on the

interval (−1, 1). We prove the existence of a minimal (strictly positive) time Tmin such that the fractionalheat dynamics can be controlled from any initial datum in L2(−1, 1) to a positive trajectory through theaction of a positive control, when s > 1/2. Moreover, we show that in this minimal time constrainedcontrollability is achieved by means of a control that belongs to a certain space of Radon measures. We also

give some numerical simulations that confirm our theoretical results.

1. Introduction

The main purpose of the present paper is to completely analysis the constrained controllablity propertiesof the heat-like equation involving the fractional Laplacian on (−1, 1). That is, the sytem

zt + (−d 2x )sz = uχω, in (−1, 1)× (0, T ), s ∈ (0, 1),

z(·, 0) = z0, in (−1, 1).(1.1)

In (1.1), the solution z is the state to be controlled and u is our control function which is localized in anopen set ω ⊂ (−1, 1).

The controllability properties of the fractional heat equation on open subsets of RN (N ≥ 2) are still notfully understood by the mathematical community. The classical tools (see e.g. [41] and the references therein)like the Carleman estimates usually used to study the controllability for heat equations are still not availablefor the fractional Laplacian (except on the whole space RN ). For this reason, our analysis in the presentarticle is limited to the one-dimensional case. Another difficulty for analyzing the system (1.1) by using somespectral properties is that contrarily to the local case s = 1 where the eigenvalues and eigenfunctions of thesystem are well known, for the fractional case, we just know an asymptotic for the eigenvalues and an explicitformula for the eigenfunction is not accessible.

In the absence of constraints, the fractional heat equation (1.1) is null-controllable in any positive timeT > 0, provided s > 1/2. This has been proved in [3] by using the gap condition on the eigenvalues, and it hasbeen validated through numerical experiments. In space dimension N ≥ 2, the best possible controllabilityresult available for the fractional heat equation is the approximate controllability recently obtained in [39].

In this work, we have obtained the following specific results:

(i) Firstly, we show that, if s > 1/2, then the system (1.1) is controllable from any given initial datumin L2(−1, 1) to zero (and, by translation, to trajectories) in any positive time T > 0 by meansof L∞-controls. This extends the analysis of [3], where only the classical case of L2-controls wasconsidered. The proof will use the canonical approach of reducing the question of controllability withan L∞-control to a dual observability problem in L1, and the use of Fourier series expansions toobtain a new result on the L1-observation of linear combinations of real exponentials.

2010 Mathematics Subject Classification. 35K05,35R11,35S05,93B05,93C20.

Key words and phrases. Fractional heat equation, constrained controllability, waiting time.This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020

research and innovation programme (grant agreement NO: 694126-DyCon). The work of the first and of the third author was

partially supported by the Grant MTM2017-92996-C2-1-R COSNET of MINECO (Spain) and by the ELKARTEK project

KK-2018/00083 ROAD2DC of the Basque Government. The work of the three authors is partially supported by the Air ForceOffice of Scientific Research under Award NO: FA9550-18-1-0242. The work of the third author was partially supported byCNCS-UEFISCDI Grant NO: PN-III-P4-ID-PCE-2016-0035 and by the Grant ICON of the French ANR.

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(ii) Secondly, as a consequence of our first result, we prove the existence of a minimal (strictly positive)time Tmin such that the fractional heat dynamics (1.1) can be controlled to positive trajectoriesthrough the action of a positive control. Moreover, if the initial datum is supposed to be positive aswell, then the maximum principle guarantees the positivity of the states too.

Fractional order operators (in particular the fractional Laplace operator) have recently emerged as amodeling alternative in various branches of science. They usually describe anomalous diffusion. A number ofstochastic models for explaining anomalous diffusion have been introduced in the literature. Among them wequote the fractional Brownian motion, the continuous time random walk, the Levy flights, the Schneidergray Brownian motion, and more generally, random walk models based on evolution equations of single anddistributed fractional order in space (see e.g. [11, 16, 26, 34]). In general, a fractional diffusion operatorcorresponds to a diverging jump length variance in the random walk.

In many PDEs models some constraints need to be imposed when considering practical applications. Thisis for instance the case of diffusion processes (heat conduction, population dynamics, etc.), where realisticmodels have to take into account that the state represents some physical quantity which must necessarilyremain positive (see, e.g., [7]).

This topic is also related to some other relevant applications, like the optimal management of compressorsin gas transportation networks, requiring the preservation of severe safety constraints (see [9, 27, 36]).

Finally, this issue is also important in other PDE problems based on scalar conservation laws, includingthe Lighthill-Whitham and Richards traffic flow models ([8, 21, 32]) or the isentropic compressible Eulerequation ([14]).

Most of the existing controllability theory for PDEs has been developed in the absence of constraintson the controls and/or state. To the best of our knowledge, the literature on constrained controllability iscurrently very limited and the majority of the available results do not guarantee that controlled trajectoriesfulfill the physical restrictions of the processes under consideration.

In the context of the heat equation, the problem of constrained controllability has been firstly addressedin [23] in the linear case, and it has been later extended to semi-linear models in [30]. In particular, in thementioned references, the authors proved that the linear and semi-linear heat equations are controllable toany positive steady state or trajectory by means of non-negative boundary controls, provided the controltime is long enough. Moreover, for positive initial data, the maximum principle guarantees that also thepositivity of the state is preserved. On the other hand, it was also proved that controllability by non-negativecontrols fails if time is too short, whenever the initial datum differs from the final target.

In addition to the results for heat-like equations, constrained controllability has been also analyzed inthe context of population dynamics. In more detail, in [17, 24] it has been shown that the controllability ofLotka-McKendrick type systems with age structuring can be obtained by preserving the positivity of thestate, once again in a long enough time horizon. These results have been recently extended in [25] to generalinfinite-dimensional systems with age structure.

The study of the controllability properties under positivity constraints is a very reasonable question forscalar-valued parabolic equations, which are canonical examples where positivity is preserved for the freedynamics. Therefore, the issue of whether the system can be controlled in between two states by means ofpositive controls, by possibly preserving also the positivity of the controlled solution, arises naturally.

The existence of a minimal time for constrained controllability is in counter-trend with respect to theunconstrained case, in which linear and semi-linear parabolic systems are known to be controllable at anypositive time. Notwithstanding, often times, norm-optimal controls allowing to reach the target at thefinal time are restrictions of solutions of the adjoint system. Accordingly these controls experience largeoscillations in the proximity of the final time, which are enhanced when the time horizon of control is small.This eventually leads to control trajectories that go beyond the physical thresholds and fail to fulfill thepositivity constraint (see [15]).

On the other hand, when the time interval is long, we expect the control property to be achieved withcontrols of small amplitude, thus ensuring small deformations of the state and, in particular, preservingits positivity. Roughly speaking, by imposing constraints to the control, we are somehow providing animpediment for the state to reach the target, unless the control time horizon is long enough. This behavior isthen a warning that existing unconstrained controllability results, that are valid within arbitrarily short time,

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may be unsuitable in practical applications in which state-constraints need to be preserved along controlledtrajectories.

In addition to the results for parabolic equations, analogous questions for the linear wave equation havebeen analyzed in [29]. There, the authors proved the controllability to steady states and trajectories throughthe action of a positive control, acting either in the interior or on the boundary of the domain considered.Nevertheless, in that case control and state positivity are not interlinked. Indeed, because of the lack of amaximum principle, the sign of the control does not determine the sign of the solution, whose positivity is nolonger guaranteed.

The rest of the paper is organized as follows. We state the main result of the paper in Section 2. In Section3, we start by presenting some preliminary technical results that are needed throughout the paper. Moreover,we give there the proof of the unconstrained controllability of the fractional heat equation (1.1) with L∞

controls. Section 4 is devoted to the proof of the main result, namely, Theorem 2.1 below. The proof isdivided in three parts. In Section 4.1, we prove the first part concerning the constrained controllability of(2.1). In Section 4.2, we obtain the strict positivity of the minimal controllability time Tmin. Section 4.3 isdevoted to the proof of the controllability in minimal time by means of measure controls. In Section 5, wepresent some numerical simulations validating our theoretical results. Finally, in Section 6, we give someconcluding remarks and propose some open problems.

2. Problem formulation and main result

In this section, we formulate precisely the problem we would like to investigate and we state our mainresults.

Let T > 0 be a real number, and define Q := (−1, 1) × (0, T ) and Qc := (−1, 1)c × (0, T ) where(−1, 1)c := R \ (−1, 1). Consider the following controllability problem for the fractional heat equation:

zt + (−d 2x )sz = uχω×(0,T ) in Q,

z = 0 in Qc,

z(·, 0) = z0(·) in (−1, 1).

(2.1)

In (2.1), ω ⊂ (−1, 1) is the control region, u is the control function and z is the state to be controlled,while for s ∈ (0, 1), the operator (−d 2

x )s is the fractional Laplacian, defined for any function v sufficientlysmooth as the following singular integral:

(−d 2x )sv(x) := cs P.V.

∫R

v(x)− v(y)

|x− y|1+2sdy, x ∈ R, (2.2)

with cs an explicit normalization constant (see e.g. [10]).It is known (see [3]) that the fractional heat equation (2.1) is null controllable in any time T > 0 by

means of a control u ∈ L2(ω × (0, T )), if and only if s > 1/2. In other words, given any z0 ∈ L2(−1, 1) andT > 0, there exists a control function u ∈ L2(ω × (0, T )) such that the corresponding unique solution z of(2.1) satisfies z(x, T ) = 0 a.e. in (−1, 1). If s ≤ 1/2, instead, null-controllability cannot be achieved and thesystem turns out to be only approximately controllable. Besides, the equation being linear, by translationthe same result holds if the final target is a trajectory z.

Moreover, it is also known (see Lemma 3.1 below) that the fractional heat equation preserves positivity.More precisely, if z0 is a given non-negative initial datum in L2(−1, 1) and u is a non-negative function, thenso it is for the solution z of (2.1). Hence, the following question arises naturally:

Can we control the fractional heat dynamics (2.1) from any initial datum z0 ∈ L2(−1, 1) toany positive trajectory z, under positivity constraints on the control and/or the state?

In other words we want to analyze whether it is possible to choose a control u ≥ 0 steering the solutionof (2.1) from z0 ∈ L2(−1, 1) to a positive trajectory z(·, T ) = z(·, T ) > 0, while possibly maintaining thissolution non-negative along the whole time interval, i.e.,

z(x, t) ≥ 0 for every (x, t) ∈ (−1, 1)× (0, T ).

Clearly, we are only interested in the case z(·, T ) 6= z0. Otherwise, the trajectory z ≡ z0 = z(·, T ) triviallysolves the problem.

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As we will see, the answer to the above question is positive, provided that the controllability time is largeenough. In particular, our main result in the present paper is the following.

Theorem 2.1. Let s > 1/2, z0 ∈ L2(−1, 1) and let z be a positive trajectory, i.e., a solution of (2.1) withinitial datum 0 < z0 ∈ L2(−1, 1) and right hand side u ∈ L∞(ω × (0, T )). Assume that there exists ν > 0such that u ≥ ν a.e in ω × (0, T ). Then, the following assertions hold.

(I) There exist T > 0 and a non-negative control u ∈ L∞(ω× (0, T )) such that the corresponding solutionz of (2.1) satisfies z(x, T ) = z(x, T ) a.e. in (−1, 1). Moreover, if z0 ≥ 0, we also have z(x, t) ≥ 0for every (x, t) ∈ (−1, 1)× (0, T ).

(II) Define the minimal controllability time by

Tmin(z0, z) := infT > 0 : ∃ 0 ≤ u ∈ L∞(ω × (0, T )) s.t. z(·, 0) = z0 and z(·, T ) = z(·, T )

. (2.3)

Then, Tmin > 0.(III) For T = Tmin, there exists a non-negative control u ∈M(ω× (0, Tmin)), the space of Radon measures

on ω × (0, Tmin), such that the corresponding solution z of (2.1) satisfies z(x, T ) = z(x, T ) a.e. in(−1, 1).

This result is in the same spirit of the ones obtained in [23, 30] in the context of the linear and semi-linearlocal heat equations under the action of a boundary control. Following the methodology presented in thementioned references, the first ingredient for proving Theorem 2.1 is to show that, in absence of constraints,(2.1) is controllable by means of an L∞-control. This will be given by the following:

Theorem 2.2. For any z0 ∈ L2(−1, 1), s > 1/2 and T > 0, there exists a control function u ∈ L∞(ω×(0, T ))such that the corresponding unique weak solution z of (2.1) with initial datum z(x, 0) = z0(x) satisfiesz(x, T ) = 0 a.e. in (−1, 1). Moreover, there is a constant C > 0 (depending only on T ) such that

‖u‖L∞(ω×(0,T )) ≤ C‖z0‖L2(−1,1). (2.4)

By means of a classical duality argument (see [12, 13, 28]), Theorem 2.2 is equivalent to the followingobservability result for the adjoint equation associated to (2.1).

Proposition 2.3. For any T > 0 and pT ∈ L2(−1, 1), let p ∈ L2((0, T );Hs0(−1, 1)) ∩ C([0, T ];L2(−1, 1))

with pt ∈ L2((0, T );H−s(−1, 1)) be the weak solution of the adjoint system−pt + (−d 2

x )sp = 0 in Q,

p = 0 in Qc,

p(·, T ) = pT (·) in (−1, 1).

(2.5)

Then, for any s > 1/2, there is a constant C = C(T ) > 0 such that

‖p(·, 0)‖2L2(−1,1) ≤ C

(∫ T

0

∫ω

|p(x, t)| dxdt

)2

. (2.6)

We refer to Section 3 for the definition of the spaces Hs0(−1, 1) and H−s(−1, 1).

We shall prove Proposition 2.3 by employing spectral techniques and with the help of the following L1

observability result for linear combinations of real exponentials.

Theorem 2.4. Let µkk≥1 be a sequence of real numbers satisfying the following conditions:

1. There exists γ > 0 such that µk+1 − µk ≥ γ for all k ≥ 1. (2.7a)

2.∑k≥1

1

µk< +∞. (2.7b)

Then, for any T > 0, there is a positive constant C = C(T ) > 0 such that, for any finite sequence ckk≥1it holds the inequality

∑k≥1

|ck|2e−2µkT ≤ C

∥∥∥∥∥∥∑k≥1

cke−µkt

∥∥∥∥∥∥2

L1(0,T )

. (2.8)

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Theorem 2.4 will follow from the classical Muntz Theorem for families of real exponentials and from theresults of [35].

We stress that Theorem 2.4, of independent interest on its own, is not specific to the equation (2.1) we areconsidering but it allows for more general results. Indeed, it yields the immediate knowledge of L∞-controlsfor one-dimensional problems simply by knowing the explicit spectrum of the equation.

3. Preliminary results

We present here some preliminary results which are needed for the proof of Theorem 2.1. We start byintroducing the appropriate function spaces needed to study our problem. For any s ∈ (0, 1) we denote by

Hs(−1, 1) :=

v ∈ L2(−1, 1) :

∫ 1

−1

∫ 1

−1

|v(x)− v(y)|2

|x− y|1+2sdxdy < +∞

the fractional order Sobolev space endowed with the norm

‖v‖Hs(−1,1) :=

(∫ 1

−1|v|2 dx+

∫ 1

−1

∫ 1

−1

|v(x)− v(y)|2

|x− y|1+2sdxdy

) 12

,

and we let

Hs0(−1, 1) :=

v ∈ Hs(R) : v = 0 on R \ (−1, 1)

.

Moreover, we let H−s(−1, 1) := (Hs0(−1, 1))? be the dual space of Hs

0(−1, 1) with respect to the pivotspace L2(−1, 1). Then we have the following continuous embeddings: Hs

0(−1, 1) → L2(−1, 1) → H−s(−1, 1).Finally, we denote by Hs

loc(−1, 1)) the space

Hsloc(−1, 1) =

v ∈ L2(−1, 1) : vϕ ∈ Hs(−1, 1) for all ϕ ∈ D(Ω)

.

If s ≥ 1, then the above spaces are defined as in [4] and their references. For more details on fractionalorder Sobolev spaces we refer to [10, 38] and their references.

We recall that according to [20, Theorem 26], for any u ∈ L2((0, T );H−s(−1, 1)) and z0 ∈ L2(−1, 1), thesystem (2.1) admits a unique weak solution

z ∈ L2((0, T );Hs0(−1, 1)) ∩ C([0, T ];L2(−1, 1)) with zt ∈ L2((0, T );H−s(−1, 1)).

Moreover, if u ∈ L2(ω × (0, T )) and z0 ≡ 0, then it has been shown in [5, Theorem 1.5] that

z ∈ L2((0, T );H2sloc(−1, 1)) ∩ L∞((0, T );Hs

0(−1, 1)) and zt ∈ L2((−1, 1)× (0, T )).

Furthermore, as we have mentioned above, the fractional heat equation preserves positivity, meaningthat, if u is non-negative and z0 is also non-negative, then the unique solution z of the system (2.1) is alsonon-negative. Such a result has been stated in [20, Theorem 26] but without giving a proof. For the sake ofcompleteness we include the full proof here. We state our result in the case N = 1 but the same holds forN ≥ 1 without any modification of the proof.

Lemma 3.1. Let u ∈ L2(ω× (0, T )) and z0 ∈ L2(−1, 1) be non-negative. Then the corresponding solution zof the system (2.1) is also non-negative.

Proof. Denote by (−d 2x )sD the realization of (−d 2

x )s in L2(−1, 1) with the zero Dirichlet exterior condition.Then, (−d 2

x )sD is the self-adjoint operator in L2(−1, 1) associated with the bilinear form E : D(E)×D(E)→ Rwith D(E) = Hs

0(−1, 1) and given by

E(ϕ,ψ) =cs2

∫R

∫R

(ϕ(x)− ϕ(y))(ψ(x)− ψ(y))

|x− y|1+2sdxdy, ϕ, ψ ∈ Hs

0(−1, 1).

We claim that (−d 2x )sD is a resolvent positive operator. Indeed, let λ > 0 be a real number, f ∈ L2(−1, 1)

and set

φ :=(λ+ (−d 2

x )sD

)−1f.

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Then, φ belongs to Hs0(−1, 1) and is a weak solution of the Dirichlet problem

(−d 2x )sφ+ λφ = f in (−1, 1),

φ = 0 in (−1, 1)c,

in the sense that

E(φ, v) + λ

∫ 1

−1φv dx =

∫ 1

−1fv dx, ∀ v ∈ Hs

0(−1, 1). (3.1)

It is clear that that there is a constant C > 0 such that

λ

∫ 1

−1|v|2 dx+ E(v, v) ≥ C‖v‖2Hs

0 (−1,1), (3.2)

for all v ∈ Hs0(−1, 1).

Now, assume that f ≤ 0 a.e. in (−1, 1) and define φ+ := maxφ, 0. It follows from [38] that φ+ ∈Hs

0(−1, 1). Let φ− := max−φ, 0. Since(φ−(x)− φ−(y)

)(φ+(x)− φ+(y)

)=φ−(x)φ+(x)− φ−(x)φ+(y)− φ−(y)φ+(x) + φ−(y)φ+(y)

=−(φ−(x)φ+(y) + φ−(y)φ+(x)

)≤ 0,

we have that E(φ−, φ+) ≤ 0. Hence,

E(φ, φ+) = E(φ+ − φ−, φ+) = E(φ+, φ+)− E(φ−, φ+) ≥ 0.

Then by (3.1), we obtain that

0 ≤ λ∫ 1

−1φφ+ dx+ E(φ, φ+) =

∫ 1

−1fφ+ dx ≤ 0.

By (3.2), the preceding estimate implies that φ+ = 0, that is, φ ≤ 0 almost everywhere. We have shownthat the resolvent (λ+ (−d 2

x )sD)−1 is a positive operator. Now it follows from the corresponding result onabstract Cauchy problems [2, Theorem 3.11.11] that, if u ≥ 0 and z0 ≥ 0, then the solution z of the fractionalheat equation (2.1) is also non-negative. We notice that this can be also seen from the representation of thesolution z. More precisely, we have that the solution z of (2.1) is given by

z(x, t) = (T (t)z0)(x) +

∫ t

0

T (t− τ)u(x, τ) dτ, (3.3)

where (T (t))t≥0 is the submarkovian (positivity-preserving and L∞-contractive) semigroup on L2(−1, 1)generated by −(−d2x)sD. The proof is finished.

We can now prove Theorem 2.2 ensuring that, without imposing any constraint on the control, it ispossible to obtain the null-controllability of (2.1) by means of a control u ∈ L∞(ω × (0, T )).

To this end, we shall first give the proof of Theorem 2.4, and then use this result to obtain the observabilityinequality (2.6).

Proof of Theorem 2.4. The proof of (2.8) is based on some results presented in [35]. Let us define thefunction

F (t) :=∑k≥1

cke−µkt.

According to [35, Section 8, page 28, Equation (8.i)] and [35, Section 9, page 33, Theorem I], under thehypothesis (2.7a) and (2.7b) the following estimates hold:

|ck| ≤ C‖F‖L1(0,T ), (3.4)∑k≥1

|ck|e(µ1−µk)t ≤ C(t)‖F‖L1(0,T ), (3.5)

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with C(t) a positive constant, depending only on t and uniformly bounded for all t > 0. Then from (3.4) and(3.5), the estimate (2.8) is immediately obtained as follows:∑

k≥1

|ck|2e−2µkT =∑k≥1

|ck|e(µ1−µk)t(|ck|e(µk−µ1)te−2µkT

)

≤ C‖F‖L1(0,T )

∑k≥1

|ck|e(µ1−µk)t ≤ C‖F‖2L1(0,T ) = C

∥∥∥∥∥∥∑k≥1

cke−µkt

∥∥∥∥∥∥2

L1(0,T )

.

The proof is finished.

We can no employ (2.8) to prove Proposition 2.3. To this end, we will first need the following technicalresult.

Lemma 3.2. Consider the eigenvalue problem for the Dirichlet fractional Laplacian in (−1, 1):(−d 2

x )sφk = λkφk, in (−1, 1)

φk = 0, in (−1, 1)c.(3.6)

That is, φkk∈N is the orthonormal basis of eigenfunctions of the operator (−d 2x )sD defined in the proof

of Lemma 3.1 with associated eigenvalues λkk∈N.Then for any open set ω ⊂ (−1, 1), there is a constantβ > 0 such that

‖φk‖L1(ω) ≥ β > 0. (3.7)

Proof. The proof is based on the asymptotic results on the spectrum of the fractional Laplacian contained inthe papers [18, 19]. Let us introduce the following auxiliary function:

q(x) :=

0 x ∈(−∞,− 1

3

),

9

2

(x+

1

3

)2

x ∈(− 1

3 , 0),

1− 9

2

(x− 1

3

)2

x ∈(0, 13),

1 x ∈(13 ,+∞

).

(3.8)

Figure 1. Graphic of the function q(x)

Moreover, for any α > 0, let us consider the function

Fα(x) = F (αx) := sin

(αx+

(1− s)π4

)−G(αx), (3.9)

7

where G is the Laplace transform of the function

γ(y) :=

√4s sin(sπ)

2π

y2s

1 + y4s − 2y2s cos(sπ)exp

(1

π

∫ +∞

0

1

1 + r2log

(1− r2sy2s

1− r2y2

)dr

).

According to [18], G is a completely monotone function satisfying

G(ξ) ≤ C(1− s)√s

ξ−1−2s, for all ξ ∈ (0,+∞). (3.10)

Then, if we define

µk :=kπ

2− (1− s)π

4, k ≥ 1, (3.11)

according to [18, Example 6.1] we have that Fµkis the solution to the equation

(−d 2x )sFµk

(x) = µkFµk(x) x > 0,

Fµk(x) = 0 x ≤ 0.

In other words, Fµkk≥1 are the eigenfunctions of the fractional Laplacian on the half-line with the zero

Dirichlet exterior condition, and µkk≥1 are the corresponding eigenvalues.Let us now define

%k(x) := q(−x)Fµk(1 + x) + (−1)kFµk

(1− x), k ≥ 1. (3.12)

Notice that Fµk(1 + x) = 0 for x ≤ −1 and Fµk

(1 − x) = 0 for x ≥ 1. From this fact, and from thedefinition (3.8) of the function q, it immediately follows that, for all k ≥ 1, %k(x) = 0 for x ∈ (−1, 1)c. Inaddition, it has been proved in [19] that there is a constant C > 0 such that∣∣(−d 2

x )s%k(x)− µ2sk %k(x)

∣∣ ≤ C(1− s)√s

µ−1k , for all x ∈ (−1, 1), k ≥ 1.

In particular, the family (%k(x), µ2sk )k≥1 is at a distance O(1/k) from the spectrum (φk(x), λk)k≥1.

Hence, instead of looking for L1-estimates of the eigenfunctions φk, we can consider the functions %k definedin (3.12).

First of all, we can trivially check that

%k(x) = Fµk(1 + x) + q(x)f(x)− sin

(µk(1 + x) +

(1− s)π4

)χ[1,+∞),

where we have denoted

f(x) := G(µk(1 + x)

)+ (−1)kG

(µk(1− x)

).

Moreover, since the open set ω ⊂ (−1, 1), then for all x ∈ ω, we have that 1± x > 0. Therefore,

supx∈ω|q(x)f(x)| ≤ c(1− s)√

sµ−1−2sk . (3.13)

In addition, for x ∈ ω we have

%k(x) = Fµk(1 + x) + q(x)f(x) = sin

(µk(1 + x) +

(1− s)π4

)−G

(µk(1 + x)

)+ q(x)f(x). (3.14)

From here, if we denote bk :=∫ω|%k(x)| dx, we can easily show that

B := infk≥1

bk > 0.

Indeed, from (3.10), (3.13) and (3.14) we have that

bk ≥∫ω

∣∣∣∣sin(µk(1 + x) +(1− s)π

4

)− c(1− s)√

sµ−1−2sk

∣∣∣∣ dx.Moreover, since limk→+∞ µ−1−2sk = 0, we have that there exists k0 ∈ N such that

bk ≥∫ω

∣∣∣∣12 sin

(µk(1 + x) +

(1− s)π4

)∣∣∣∣ dx, for all k > k0.

8

It follows that B1 := infk>k0 bk > 0. Hence, B > 0 since bk > 0 for all k (the integrand being positiveexcept possibly for a set of zero measure in which it is zero). The proof is then concluded.

Remark 3.3. We mention that it has been shown in [6, Equation (5.3)] that the first positive eigenfunction

φ1 of the fractional Laplacian satisfies φ1(x) '(dist(x, (−1, 1)c)

)s= (1− |x|)s, in the sense that, there are

two positive constants 0 < C1 ≤ C2 such that

C1(1− |x|)s ≤ φ1(x) ≤ C2(1− |x|)s, x ∈ (−1, 1). (3.15)

Proof of Proposition 2.3. The proof of the estimate (2.6) is based on standard spectral techniques. Firstof all, it is immediate to show that the solution of (2.5) can be expressed in the basis of the eigenfunctionsφkk≥1 of the operator (−d2x)sD as follows:

p(x, t) =∑k≥1

akφk(x)e−λk(T−t), ak :=

∫ 1

−1pT (x)φk(x) dx. (3.16)

From (3.16), by using the orthonormality of the eigenfunctions in L2(−1, 1) and the change of variablesT − t 7→ t, it is easy to see that the observability inequality (2.6) can be rewritten as

∑k≥1

|ak|2e−2λkT ≤ C

∫ T

0

∫ω

∣∣∣∣∑k≥1

akφk(x)e−λkt

∣∣∣∣ dxdt2

.

Moreover, employing the lower bound (3.7) for the L1(ω)-norm of φk, we immediately get∫ T

0

∫ω

∣∣∣∣∑k≥1

akφk(x)e−λkt

∣∣∣∣ dxdt ≥ β ∫ T

0

∣∣∣∣∑k≥1

ake−λkt

∣∣∣∣ dt = β

∥∥∥∥∥∥∑k≥1

ake−λkt

∥∥∥∥∥∥L1(0,T )

.

Hence, in order to obtain (2.6) it will be enough to show that it holds the inequality

∑k≥1

|ak|2e−2λkT ≤ C

∥∥∥∥∥∥∑k≥1

ake−λkt

∥∥∥∥∥∥2

L1(0,T )

. (3.17)

Notice that (3.17) is nothing more than (2.8) with ck = ak and µk = λk. Hence, we only need to showthat the eigenvalues of the fractional Laplacian satisfy (2.7a) and (2.7b).

For the gap condition (2.7a), we can use [19, Proposition 3] ensuring that the eigenvalues λk are all simpleif s ≥ 1/2.

Concerning instead the summability in (2.7b), according to [19, Theorem 1] we have the followingasymptotic behavior:

λk =

(kπ

2+

(1− s)π4

)2s

+O(

1

k

)as k →∞. (3.18)

Hence, by means of (3.18), it is immediate to see that (2.7b) holds if and only if s > 1/2. If s ≤ 1/2,instead,

∑k≥1 λ

−1k behaves as the harmonic series and therefore, it is divergent.

Summarizing, since we are assuming s > 1/2, we have that the two conditions (2.7a) and (2.7b) are bothsatisfied and, consequently, (3.17) holds. The proof is finished.

Proof of Theorem 2.2. The proof is based on a duality argument. This approach being classical in PDEcontrol theory, for brevity we are going to present here only the principal ideas. The interested reader mayfound the complete details in [28].

Let us fix T > 0. For every pT ∈ L2(−1, 1), let p ∈ L2((0, T );Hs0(−1, 1)) ∩ C([0, T ];L2(−1, 1)) with

pt ∈ L2((0, T );H−s(−1, 1)) be the unique weak solution of the adjoint system (2.5).Then, according to [28, Proposition 2.6], for any z0 ∈ L2(−1, 1) the corresponding solution of (2.1) is null

controllable at time T by means of a control function u ∈ L∞(ω × (0, T )) if and only if the observabilityinequality (2.6) holds. Moreover, following step by step the proof of that proposition it is easy to obtain

9

the inequality (2.4) for the L∞-norm of the control. Finally, according to [28, Proposition 2.7], the controlfunction u is such that

‖u‖L∞(ω×(0,T )) = ‖p‖L1(ω×(0,T )). (3.19)

The proof is then concluded.

4. Proof of the main result

In this section, we give the proof of our main result, namely, Theorem 2.1. The proof will be divided inthree parts.

4.1. Constrained controllability of the system (2.1). We present here the proof of the first part ofTheorem 2.1 concerning the controllability of (2.1) through a non-negative control u ∈ L∞(ω × (0, T )).

Proof of Theorem 2.1(I). First of all, observe that, since the equation is linear, by subtracting z, it isenough to show that there exist a time T > 0 and a control v ∈ L∞(ω × (0, T )) fulfilling the constraintv > −ν a.e. in ω × (0, T ), such that the solution ξ of the system

ξt + (−d 2x )sξ = vχω×(0,T ) in Q,

ξ = 0 in Qc,

ξ(·, 0) = z0(·)− z0(·) in (−1, 1),

(4.1)

satisfies ξ(x, T ) = 0 a.e. in (−1, 1).According to Proposition 2.3, the null-controllability of (4.1) with v ∈ L∞(ω × (0, T )) is equivalent to the

observability inequality (2.6). Actually, the controllability (and therefore the observability of the adjointsystem) being true for any time interval (τ, T ), we also have

‖p(·, τ)‖2L2(−1,1) ≤ C(T − τ)

(∫ T

τ

∫ω

|p(x, t)| dxdt

)2

.

Using (3.16), the fact that the eigenvalues λkk≥1 of the operator (−d2x)sD form a non-decreasing sequence,and the dissipativity of the fractional heat semigroup ensuring exponential stability, we can readily checkthat

‖p(·, 0)‖2L2(−1,1) ≤ e−2λ1τ‖p(·, τ)‖2L2(−1,1)

for every 0 < τ < T and therefore,

‖p(·, 0)‖2L2(−1,1) ≤ e−2λ1τC(T − τ)

(∫ T

0

∫ω

|p(x, t)| dxdt

)2

.

By duality, this means that the control v can be chosen such that

‖v‖2L∞(ω×(0,T )) ≤ e−2λ1τC(T − τ)‖z0 − z0‖2L2(−1,1).

Taking τ = T/2, we obtain

‖v‖2L∞(ω×(0,T )) ≤ e−λ1TC(T )‖z0 − z0‖2L2(−1,1).

Furthermore, we recall that the observability constant C(T ) is uniformly bounded for any T > 0. Hence,for T large enough we have

‖v‖2L∞(ω×(0,T )) < ν.

This immediately implies that v > −ν. Therefore, we have shown the existence of a control v > −νsteering the solution of (4.1) from z0 − z0 to zero in time T > 0, provided T is large enough. Consequently,the solution z of (2.1) is controllable to the trajectory z in time T .

Finally, if z0 ≥ 0, thanks to Lemma 3.1 we also have z(x, t) ≥ 0 for every (x, t) ∈ (−1, 1)× (0, T ). Theproof is finished.

10

4.2. Positivity of the minimal time for constrained controllability. This section is devoted to theproof of the second part of Theorem 2.1 that shows that the minimal time Tmin needed for controlling thesystem (2.1) with a non-negative control u ∈ L∞(ω × (0, T )) is necessarily strictly positive.

Proof of Theorem 2.1(II). Let us start by writing the solution of (2.1) in the basis of the eigenfunctionsφkk≥1, that is,

z(x, t) =∑k≥1

zk(t)φk(x), (4.2)

with

zk(t) :=

∫ 1

−1z(x, t)φk(x) dx. (4.3)

Derivating (4.3) and using (2.1) (or multiplying (2.1) by φk and integrating over (−1, 1)), we can readilycheck that the coefficients zk(t) satisfy the following first order ODE:

z′k(t) = −λkzk(t) + uk(t), t ∈ (0, T )

zk(0) =

∫ 1

−1z0(x)φk(x) dx =: z0k,

where we have denoted

uk(t) :=

∫ω

u(x, t)φk(x) dx.

Hence, employing the variation of constants formula we easily get that

zk(t) = z0ke−λkt +

∫ t

0

e−λk(t−τ)uk(τ) dτ. (4.4)

On the other hand, since z(x, T ) = z(x, T ) a.e. in (−1, 1), we have that

zk(T ) =

∫ 1

−1z(x, T )φk(x) dx =: ζk,

and from (4.4) we immediately obtain that

ζk − z0ke−λkT =

∫ T

0

e−λk(T−τ)uk(τ) dτ.

Now, for every 0 ≤ τ ≤ T , we have

e−λkT ≤ e−λk(T−τ) ≤ 1.

Therefore (notice that uk ≥ 0),

e−λkT

∫ T

0

uk(τ) dτ ≤∫ T

0

e−λk(T−τ)uk(τ) dτ ≤∫ T

0

uk(τ) dτ,

from which we obtain that

ζk − z0ke−λkT ≤∫ T

0

uk(τ) dτ ≤ ζkeλkT − z0k. (4.5)

Assume by contradiction that, for every T > 0, there exists a non-negative control function uT steeringz0 to z(·, T ) in time T , and that z(·, T ) 6= z0 (otherwise, the trajectory z ≡ z0 = z(·, T ) trivially solves theproblem). Then, (4.5) ensures that

limT→0+

∫ T

0

uTk (τ) dτ = ζk − z0k =: γ =⇒ z0k = ζk − γ.

Since z0 ∈ L2(−1, 1), we clearly have∑k≥1

|z0k|2 =∑k≥1

(ζ2k − 2γζk + γ2

)< +∞,

11

which implies that

limk→+∞

(ζ2k − 2γζk + γ2

)= 0. (4.6)

Moreover, since φkk≥1 is an orthonormal complete system in L2(−1, 1), it follows that φk 0 (weakconvergence) in L2(−1, 1) as k → +∞. This implies that

limk→+∞

(z(·, T ), φk)L2(−1,1) = limk→+∞

∫ 1

−1z(x, T )φk(x) dx = lim

k→+∞ζk = 0.

This identity and (4.6) yield γ = 0. Thus, we immediately have that

0 = z0k − ζk =

∫ 1

−1(z0(x)− z(x, T ))φk(x) dx, for all k ≥ 1.

This is possible if and only if z0(x) = z(x, T ) a.e. in (−1, 1), which contradicts our previous assumption.The proof is then concluded.

4.3. Constrained controllability in minimal time with measure controls. In this section, we givethe proof of the third part of Theorem 2.1 which ensures that constrained controllability of the system (2.1)holds in the minimal time Tmin with controls in the (Banach) space of the Radon measures M(ω × (0, Tmin))endowed with the norm

‖µ‖M(ω×(0,Tmin)) = sup

∫ω×(0,Tmin)

ϕ(x, t) dµ(x, t) : ϕ ∈ C(ω × [0, Tmin],R), maxω×[0,Tmin]

|ϕ| = 1

.

We recall that solutions of (2.1) with controls in M(ω × (0, Tmin)) are defined by transposition (see [22]).

Definition 4.1. Given z0 ∈ L2(−1, 1), T > 0, and u ∈ M(ω × (0, T )), we shall say that the functionz ∈ L1(Q), is a solution of (2.1) defined by transposition if it satisfies the identity∫

ω×(0,T )

p(x, t) du(x, t) = 〈z(·, T ), pT 〉 −∫ 1

−1z0(x)p(x, 0) dx, (4.7)

where, for every pT ∈ L∞(−1, 1), the function p ∈ L∞(Q) is the unique solution of the adjoint system−pt + (−d 2

x )sp = 0 in Q,

p = 0 in Qc,

p(·, T ) = pT in (−1, 1).

(4.8)

The existence of a unique transposition solution z ∈ L1(Q) of (2.1) is a consequence of the maximumprinciple for parabolic equations together with duality and approximation arguments. These arguments beingclassical, we omit here the technical details.

Proof of Theorem 2.1(III). Let us now prove the existence of a measure-valued non-negative controlu ∈M(ω × (0, Tmin)) realizing the controllability of the system (2.1) exactly in time Tmin. Let us denote

Tk := Tmin +1

k, k ≥ 1.

In view of the definition (2.3) of the minimal control time, there exists a sequence of non-negative controlsuTkk≥1 ⊂ L∞(ω × (0, Tk)) such that the corresponding solution zk of (2.1) with zk(x, 0) = z0(x) a.e. in(−1, 1) satisfies zk(x, Tk) = z(x, Tk) a.e. in (−1, 1). We extend these controls by u on (Tk, Tmin + 1) to get anew sequence (that we shall denote again by uTk) in L∞(ω × (0, Tmin+1)).

We want to prove that this sequence is bounded in L1(ω × (0, Tmin+1)). To this end, let us assume thatthe initial datum pT in (4.8) is positive which, thanks to Lemma 3.1, implies that the corresponding solution

12

p satisfies p(x, t) ≥ θ > 0 for all (x, t) ∈ (−1, 1)× (0, Tmin + 1) and for some positive constant θ. Then, (4.7)and the positivity of uTk yield

θ∥∥uTk

∥∥L1(ω×(0,Tmin+1))

= θ

∫ Tmin+1

0

∫ω

uTk(x, t) dxdt ≤∫ Tmin+1

0

∫ 1

−1p(x, t)uTk(x, t) dxdt

= 〈z(·, T ), pT 〉 −∫ 1

−1z0(x)p(x, 0) dx ≤M,

where the last inequality is due to the continuous dependence of the solutions of the direct and adjointproblems on the initial data.

We have shown that the sequence uTkk≥1 is bounded in L1(ω× (0, Tmin+1)), hence, it is bounded in thespace M(ω × (0, Tmin+1)). Thus, extracting a sub-sequence if necessary, we have that

uTk∗ u weakly-∗ in M(ω × (0, Tmin+1)) as k → +∞.

Clearly, the limit control u satisfies the non-negativity constraint.Now, for any k large enough and Tmin < T0 < Tmin+1, by (4.7) and the definition of the control uTk we

have ∫ω×(0,T0)

p(x, t) duTk(x, t) = 〈z(·, T0), pT 〉 −∫ 1

−1z0(x)p(x, 0) dx. (4.9)

Letting pT be the first non-negative eigenfunction φ1 of (−d2x)sD (see (3.15)), or generally any non-negativefunction in the domain D((−d2x)sD) of the operator (−d2x)sD, we get that the solution p of the system (4.8)belongs to C([0, T ];D((−d2x)sD)) → C([0, T ] × [−1, 1]). Therefore, by definition of weak∗ limit, lettingk → +∞ in (4.9) we obtain∫

ω×(0,T0)

p(x, t) d u(x, t) = 〈z(·, T0), pT 〉 −∫ 1

−1z0(x)p(x, 0) dx,

which in turn implies that z(x, T0) = z(x, T0) a.e. in (−1, 1). Finally, taking the limit as T0 → Tmin andusing the fact that

|u|(ω × (Tmin, T0)) = |u|(ω × (Tmin, T0)) = 0, as T0 → Tmin,

we can deduce that z(x, Tmin) = z(x, Tmin) a.e. in (−1, 1). This concludes the proof.

4.4. Lower bounds for the minimal controllability time. Theorem 2.1 shows that the fractional heatequation (2.1) is controllable to positive trajectories by means of a non-negative control u, provided that thecontrollability time is large enough. Moreover, in the minimal controllability time Tmin defined by (2.3), weproved that the non-negative controls are in the spaceM(ω× (0, Tmin)) of Radon measures. Notwithstanding,our result does not provide a precise estimate for Tmin.

This is indeed a delicate issue. In [30], it has been addressed for the case of the classical linear andsemi-linear heat equations by means of a quite general approach. Nevertheless, this methodology does notimmediately apply to the fractional heat equation (2.1).

In order to clarify this point, in what follows we present an abridged description of how the techniquesdeveloped in [30] should be applied in our case and we highlight the difficulties we encounter.

The starting point is to notice that, by a simple translation argument, we can reduce ourselves to considerthe case in which the fractional heat equation (2.1) has zero initial datum. Then, from the definition oftransposition solution (4.7) we have

〈z(·, T ), pT 〉 −∫ω×(0,T )

p(x, t) du(x, t) = 0.

Following the procedure of [30], the idea is now to find T0 > 0 and pT ∈ L2(−1, 1) such that thecorresponding solution of the adjoint system (2.5) satisfies

p ≥ 0, in ω × (0, T0),

〈 z(·, T ), pT 〉 < 0, for all T ∈ [0, T0).(4.10)

13

Then, an explicit lower bound of the controllability minimal time is obtained by analyzing sharply theconditions required for (4.10) to hold. See [30, Sections 5.1 and 6.1] for more details.

The choice of a suitable initial datum for the adjoint equation (2.5) is not at all obvious. In [30], in thecase of the linear and semi-linear heat equations with boundary control, the authors propose to consider pTin the form

pT = −φ1 + 2(1− ζ)φ1 (4.11)

or

pT = −αφ1 + βφ3, (4.12)

where φ1 and φ3 are respectively the first and third eigenfunction of the Dirichlet Laplacian, α and β aresuitable positive constants, and ζ is a cut-off function supported outside the control region.

With these choices, a lower estimate for Tmin is obtained by employing the positivity of φ1 and the explicitknowledge of the eigenfunctions φ1 and φ3.

Nevertheless, the above proposals for pT do not seem to be appropriate for our fractional heat equation(2.1). This for at least two main reasons. On the one hand, we cannot ensure that with pT in the form(4.11) or (4.12) the corresponding solution of the adjoint equation (2.5) remains positive in ω. On the otherhand, even if we were able to overcome this first difficulty, for the eigenfunctions of the Dirichlet fractionalLaplacian we do not have a nice expression in terms of sinusoidal functions, as in the case of the classicallocal operator. Therefore, to perform explicit estimates is a much more difficult issue.

In view of this discussion, we can conclude that the methodology just presented is not immediatelyapplicable to the context of the present paper. For this reason, we are not able to provide explicit analyticlower estimates for the minimal time. This will be done, instead, in Section 5 through a numerical approach.

5. Numerical simulations

Our main result, Theorem 2.1, states that the fractional heat equation (2.1) is controllable from anyinitial datum z0 ∈ L2(−1, 1) to any positive trajectory z by means of the action of a non-negative controlu ∈ L∞(ω × (0, T )), provided that s > 1/2 and the controllability time T is large enough.

In this section, we present some numerical simulations that confirm these theoretical results. In particular,we will focus on two specific situations:

• Case 1: we choose as initial datum

z0(x) = 2 cos(π

2x),

and we set as a target z(·, T ) the solution at time T of (2.1) with initial datum

z0(x) =1

20cos(π

2x)

and right-hand side u ≡ 1/5. In this case, as it can be observed in figure 2, we have z0 > z(·, T ) a.e.in (−1, 1).

• Case 2: we choose as initial datum

z0(x) =1

2cos(π

2x),

and we set as a target z(·, T ) the solution at time T of (2.1) with initial datum

z0(x) = 6 cos(π

2x)

and right-hand side u ≡ 1. In this case, as it can be observed in figure 8, we have z0 < z(·, T ) a.e. in(−1, 1).

In both cases, we first estimate numerically Tmin by formulating the minimal-time control problem as anoptimization one. In a second moment, we will show that for T ≥ Tmin the fractional heat equation (2.1)is controllable from z0 ∈ L2(−1, 1) to the given trajectories z(·, T ) (see above) by means of a non-negativecontrol u, while for T < Tmin this controllability result is not achieved.

To simplify the presentation, we always choose the interval ω = (−0.3, 0.8) ⊂ (−1, 1) as the control region.Moreover, we focus on the case s > 1/2, in which we know that (2.1) is controllable. We recall that, if

14

s ≤ 1/2, it has been shown in [3], both on a theoretical and numerical level, that the fractional heat equation(2.1) is not controllable, not even in the absence of constraints.

5.1. Case 1: z0 > z(·, T ). Let us start by analyzing the case of an initial datum z0 above the finaltarget z(·, T ). As we mentioned, we first estimate the minimal controllability time Tmin through a suitableoptimization problem and we address the numerical constrained controllability of (2.1) in a time horizonT ≥ Tmin. In a second moment, we will consider the case T < Tmin.

5.1.1. Numerical approximation of the minimal controllability time. To obtain an approximation of theminimal controllability time Tmin, we solve the following constrained optimization problem:

minimize T (5.1)

subject to

T > 0,

zt + (−d 2x )sz = uχω, a.e. in (−1, 1)× (0, T ),

z(x, 0) = z0 ≥ 0, a.e. in (−1, 1), (5.2)

z(x, t) ≥ 0, a.e. in (−1, 1)× (0, T ),

u(x, t) ≥ 0, a.e. in ω × (0, T ).

To solve this problem numerically, we employ the expert interior-point optimization routine IpOpt (see[37]) combined with automatic differentiation and the modeling language AMPL ([31]).

To perform our simulations, we apply a FE method for the space discretization of the fractional Laplacianon a uniform space-grid xi = −1 + 2i

Nx, i = 1, . . . , Nx, with Nx = 20 (see [3]). Moreover, we use an explicit

Euler scheme for the time integration on the time-grid tj = TjNt

, j = 0, . . . , Nt, with Nt satisfying theCourant-Friedrich-Lewy condition. In particular, we will choose here Nt = 300.

The minimal time that we obtain from our simulations is Tmin ' 0.8285 and we can see in figure 2 that, inthis time horizon, we are able to steer the initial datum z0 to the desired target by maintaining the positivityof the solution.

Figure 2. Evolution in the time interval (0, Tmin) of the solution of (2.1) with s = 0.8.The blue curve is the target we want to reach while the green bullets indicate the target wecomputed numerically.

We have to stress here that the minimal time Tmin we have obtained is just an approximation computedby solving numerically the optimization problem (5.1), (5.2). The validity of this approximation shall beconfirmed by a convergence result as the mesh-sizes tend to zero. We will present more details on this issuein Section 6.

In figures 3 and 4, we show the behavior of the minimal-time control corresponding to the dynamics offigure 2. As we can see, the control is initially inactive and leaves the equation evolving under the dissipativeeffect of the heat semigroup. When the state finally approaches the target, the control prevents it to pass

15

below by means of an impulsional action localized in certain specif points of the control region. Moreover,we have to mention that, since the range of amplitudes of the impulses of our minimal-time control is quitelarge, the plot in figure 4 (and in figure 10 below) is in logarithmic scale. In this way, also the impulses withsmaller amplitude can be appreciated.

Figure 3. Minimal-time control: space-time distribution of the impulses. The white linesdelimit the control region ω = (−0.3, 0.8). The regions in which the control is active aremarked in yellow.

Figure 4. Minimal-time control: intensity of the impulses in logarithmic scale. In the (t, x)plane in blue the time t varies from t = 0 (left) to t = Tmin (right).

This behavior of the control is not surprising. Indeed, as it was already observed in [23, 30], the minimal-time controls are expected to be atomic measures, in particular linear combinations of Dirac deltas. Oursimulations are thus consistent with the aforementioned papers. Nevertheless, a more complete analysis ofthe positions and amplitudes of these impulses shall be addressed. This will be the subject of a future work.

The impulsional behavior of the control is then lost when extending the time horizon beyond Tmin. Infigure 5, we show the evolution of the solution of the fractional heat equation (2.1) from the initial datumz0 to the target z(·, T ) in the time horizon T = 0.9. As we can observe, in accordance with our theoreticalresults, the equation is still controllable in time T . Nevertheless, the action of the control is now moredistributed in ω (see figure 6).

5.1.2. Lack of controllability in short time. In this section, we conclude our discussion on Case 1 by showingthe lack of controllability of (2.1) when the time horizon is too short.

To this end, we employ a classical conjugate gradient method implemented in the DyCon ComputationalToolbox ([1]) for solving the following optimization problem:

min ‖z(·, T )− z(·, T )‖L2(−1,1) (5.3)

16

Figure 5. Evolution in the time interval (0, 0.9) of the solution of (2.1) with s = 0.8. Theblue curve is the target we want to reach while the green bullets indicate the target wecomputed numerically.

Figure 6. Behavior of the control in time T = 0.9. The white lines delimit the controlregion ω = (−0.3, 0.8). The regions in which the control is active are marked in yellow. Theatomic nature is lost.

subject to

zt + (−d 2x )sz = uχω, a.e. in (−1, 1)× (0, T ),

z(x, 0) = z0 ≥ 0, a.e. in (−1, 1), (5.4)

z(x, t) ≥ 0, a.e. in (−1, 1)× (0, T ),

u(x, t) ≥ 0, a.e. in ω × (0, T ).

As before, we apply a FE method for the space discretization of the fractional Laplacian on a uniformspace-grid with Nx = 20 points and we use an explicit Euler scheme for the time integration on a time-gridwith Nt = 300 points. Furthermore, we choose a time horizon T = 0.7, which is below the minimalcontrollability time Tmin.

Our simulation then show that the solution of (2.1) fails to be controlled (see figure 7). In fact, since wewant to reach a final target which is below the initial datum z0, the natural approach would be to pushdown the state with a ‘‘negative’’ action. Since, however, the control is not allowed to do this because ofthe non-negativity constraint, its best option is to remain inactive for the entire time interval and to let thesolution diffuses under the action of the fractional heat semigroup. Nevertheless, this is not enough to reachthe target in the time horizon provided.

5.2. Case 2: z0 < z(·, T ). Let us now consider the case of an initial datum z0 which is smaller than thefinal target z(·, T ). As before, we start by using IpOpt for solving the optimization problem (5.1), (5.2) andcomputing the minimal controllability time.

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Figure 7. Evolution in the time interval (0, 0.7) of the solution of (2.1) with s = 0.8 (left)and of the control u (right), under the constraint u ≥ 0. The bold characters highlight thecontrol region ω = (−0.3, 0.8). The control remains inactive during the entire time interval,and the the equation is not controllable.

Also in this case, we apply a FE method for the space discretization of the fractional Laplacian on auniform space-grid with Nx = 20 points and we use an explicit Euler scheme for the time integration on atime-grid with Nt = 100 points.

This time, we obtain Tmin ' 0, 2101 and, once again, our simulations displayed in figure 8 show that inthis time horizon the fractional heat equation (2.1) is controllable from the initial datum z0 to the desiredtrajectory z(·, T ).

Figure 8. Evolution in the time interval (0, Tmin) of the solution of (2.1) with s = 0.8.The blue curve is the target we want to reach while the green bullets indicate the target wecomputed numerically.

Nevertheless, notice that, this time, we want to reach a target which is above the initial datum z0. Thismeans that the control needs to countervail also the dissipation of the solution of (2.1), by acting on it fromthe very beginning with a positive force. Moreover, also in this case, the minimal-time control has an atomicnature, as it is shown in figures 9 and 10.

Moreover, when extending the time horizon beyond Tmin we can observe once again how the solution of(2.1) is still controlled but, this time, the control is distributed in the a larger part of the control region ωand not anymore localized in specific points (see Figures 11 and 12)

Finally, when considering a time horizon T < Tmin we can notice once more that the solution of (2.1)fails to be controlled to the desired trajectory z(·, T ). In fact, Figure 13 shows that the numerical target(displayed in green) computed by employing the tolls of the DyCon computational toolbox for solving theoptimization problems (5.3), (5.4) does not totally match the desired target in blue.

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Figure 9. Minimal-time control: space-time distribution of the impulses. The white linesdelimit the control region ω = (−0.3, 0.8). The regions in which the control is active aremarked in yellow.

Figure 10. Minimal-time control: intensity of the impulses in logarithmic scale. In the(t, x) plane in blue the time t varies from t = 0 (left) to t = Tmin (right).

Figure 11. Evolution in the time interval (0, 0.4) of the solution of (2.1) with s = 0.8. Theblue curve is the target we want to reach while the green bullets indicate the target wecomputed numerically.

6. Concluding remarks

In this paper, we have studied the controllability to trajectories for a one-dimensional fractional heatequation under nonnegativity state and control constraints.

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Figure 12. Behavior of the control in time T = 0.4. The white lines delimit the controlregion ω = (−0.3, 0.8). The regions in which the control is active are marked in yellow. Theatomic nature is lost.

Figure 13. Evolution in the time interval (0, 0.15) of the solution of (2.1) with s = 0.8(left) and of the control u (right), under the constraint u ≥ 0. The bold characters highlightthe control region ω = (−0.3, 0.8). The equation is not controllable.

For s > 1/2, when controllability for the unconstrained fractional heat equation holds in any positivetime T > 0 by means of an L2-distributed control, we have shown that the introduction of state or controlconstraints creates a positive minimal time Tmin for achieving the same result. Moreover, we have also provedthat, in this minimal time, constrained controllability holds with controls in the space of Radon measures.

Our results, which are in the same spirit of the analogous ones obtained in [23, 30] for the linear andsemi-linear heat equations in one and several space dimensions, are supported by the numerical simulationsin Section 5.

We present hereafter a non-exhaustive list of open problems and perspectives related to our work.

1. Extension to the multi-dimensional case. Our analysis is based on spectral techniques, andapplies only to a one-dimensional fractional heat equation. The extension to multi-dimensionalproblems on bounded domains Ω ⊂ RN , N ≥ 1, requires different tools such as Carleman estimates.Nevertheless, obtaining Carleman estimates for the fractional Laplacian is a very difficult issue whichhas been addressed only partially, and only for problems defined on the whole Euclidean space RN(see, e.g., [33]). The case of bounded domains remains open and it is quite challenging. As oneexpects, the main difficulties come from the non-local nature of the fractional Laplacian, which makesclassical PDEs technique more delicate or even impossible to use.

2. Bang-bang nature of the controls. In Theorem 2.2 we showed that the fractional heat equation(2.1) is controllable to trajectories by means of L∞-controls satisfying ‖u‖L∞(ω) = ‖p‖L1(ω), p being

the solution of the adjoint equation (2.5). It is then a natural and interesting question to analyze20

whether these controls have a bang-bang nature, that is,

u = ‖p‖L∞(ω×(0,T ))sign(p).

To this end, the first step would be to show that the zero set of the solutions of the adjoint equationis of null measure, so that the sign of the adjoint state is well defined. This is true in the case of theclassical heat equation, as a consequence of the space-time analyticity properties of the solutions.Nevertheless, we do not know whether the same holds also for the fractional heat equation (2.1) since,in this case, as far as we know no space-time analytic regularity results are available (in fact, it isknown that solutions are analytic in time, but the space analyticity is still an open problem). Then,this becomes a very challenging problem, both in PDE analysis and control theory.

3. Constrained controllability from the exterior for the fractional heat equation. In [40],the null-controllability from the exterior for the one-dimensional fractional heat equation has beentreated. In particular, it has been proved there that, if s > 1/2, null controllability is achievableby means of a control function g acting on a subset O ⊂ (R \ (0, 1)). Hence, to address constrainedcontrollability in this framework becomes a very interesting issue.

4. Lower bounds for the minimal constrained controllability time. In Section 5, we gave somenumerical lower bound for the minimal constrained controllability time. Nevertheless, the boundswe presented are not optimal. This raises two very important issues. On the one hand, we shallobtain analytical lower bounds for the controllability time. This question was already addressed in[23, 30] for the local heat equation but, as we discussed in Section 4.4, the methodology developed inthose works does not apply immediately to our case. Therefore, there is the necessity to adapt thetechniques of [23, 30], or to develop new ones. On the other hand, we should develop a completeanalysis of the efficiency of the numerical method we used for estimating this minimal time, in orderto determine the accuracy of our approximation.

5. Convergence result for the minimal time. The minimal time Tmin in the simulations of Section5 is just an approximation computed by solving numerically the optimization problem (5.1), (5.2).The validity of these computational result should be confirmed by showing that this minimal time ofcontrol for the discrete problem converges towards the continuous one as the mesh-sizes tend to zero.This could be done by adapting the procedure presented in [23, Section 5.3]. Nevertheless, we haveto mention that, in order to corroborate this procedure, it is required the knowledge of an analyticlower bound for Tmin which, at the present stage, it is unknown (see point 4 above).

Acknowledgments

The authors want to thank Debayan Maity (DeustoTech, University of Deusto, Bilbao, Spain) for hisvaluable help in some delicate technical issues of our proofs. A special thanks goes to Jesus Oroya (DeustoTech,University of Deusto, Bilbao, Spain) for his contribution to the simulations in Section 5.

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1 DeustoTech, University of Deusto, 48007 Bilbao, Basque Country, Spain.

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2 Facultad de Ingenierıa, Universidad de Deusto, Avenida de las Universidades 24, 48007 Bilbao, Basque Country,Spain, +34 944139003 - 3282.

Email address: umberto.biccari@deusto.es, u.biccari@gmail.com

3 University of Puerto Rico, Rio Piedras Campus, Department of Mathematics, College of Natural Sciences, 17

University AVE. STE 1701 San Juan PR 00925-2537 (USA)

Email address: mahamadi.warma1@upr.edu

4 Departamento de Matematicas, Universidad Autonoma de Madrid, 28049 Madrid, Spain.

5 Sorbonne Universites, UPMC Univ Paris 06, CNRS UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris,France.

Email address: enrique.zuazua@deusto.es

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